有限域上点到平面关联的组合方法

中文摘要 AI 辅助

我们在PG(n+d,q)中建立了点集P与n-平面族L之间关联的新界。对于固定维度，我们的关联偏差界具有关于$\log_q|L|$的显式分段线性指数，在平面数量的特定范围内改进了Haemers的经典估计和Kong-Tamo界。匹配构造在若干参数范围内证明了至多常数因子的最优性。证明是组合性的，避免了谱和傅里叶分析方法。作为应用，我们获得了丰富平面和例外正交投影的改进估计，以及在规定方向比例较小的某些范围内Furstenberg集的更强下界。

英文摘要

We establish new bounds for incidences between a point set P and a family L of n-flats in PG(n+d,q). For fixed dimensions, our bound on the incidence discrepancy has an explicit piecewise-linear exponent in $\log_q|L|$, improving the classical estimate of Haemers and the Kong-Tamo bound in specified ranges of the number of flats. Matching constructions establish sharpness up to constant factors in several parameter ranges. The proof is combinatorial and avoids spectral and Fourier analytic methods. As applications, we obtain improved estimates for rich flats and exceptional orthogonal projections, together with stronger lower bounds for Furstenberg sets in certain ranges where the fraction of prescribed directions is small.

发表机构

• Institute of Mathematics and Interdisciplinary Sciences（数学与交叉科学研究院）
• Xidian University（西安电子科技大学）

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